Place whole potatoes in a heavy pot with lightly salted water to cover and bring to a boil. Lower heat to a low boil and cook until just tender, about 45 minutes. When cool enough to handle, peel and force through ricer or food mill onto a sheet pan, spreading in an even layer. Cool potatoes completely.
On a lightly floured board, gather potatoes into a mound in sheet pan, using a pastry scraper if you have one, and form a well in center. Pour beaten eggs and salt into the well, then blend into the potatoes. Sprinkle 1 1/2 cups of the flour over the potatoes and gently mix in. Add remaining flour as needed until a dough just forms. Dust top lightly with some flour and refrigerate until ready to use.
Bring water to a boil with 1-2 tablespoons of salt. Pinch off a walnut-sized piece of dough and flatten into a 2 1/2 inch circle. Place a quarter plum piece stuffed with a half a sugar cube in the center of each round of dough. Close the dough around the filling to form a smooth package with no tears. Repeat with remaining dough and plums.
Melt butter in a large sauté pan over medium heat. Add the bread crumbs and toast until golden brown. Remove from heat and set aside.
Add gnocchi, 8 or so at a time (so that they don’t touch) to the boiling water, stirring occasionally to prevent sticking. Wait for them to rise to the top and continue to cook an additional 2 minutes.
Remove with a slotted spoon and drain well. Add to the bread crumb mixture and gently toss to coat.
Place on a serving platter and sprinkle with remaining bread crumbs, cinnamon and sugar to taste. Serve warm as an appetizer, pasta main or dessert.